In this study, methodologies for modeling grey uncertainty in water resources systems are developed, specifically for the problems in two identified areas in water resources: waste load allocation in streams and floodplain planning. A water resources system is associated with some degree of uncertainty, due to randomness of hydrologic and hydraulic parameters, imprecision and subjectivity in management goals, inappropriateness in model selection, inexactness of different input parameters for inadequacy of data, etc. Uncertainty due to randomness of input parameters could be modeled by the probabilistic models, when probability distributions of the parameters may be estimated. Uncertainties due to imprecision in the management problem may be addressed by the fuzzy decision models. In addition, some parameters in any water resources problems need to be addressed as grey parameters, due to inadequate data for an accurate estimation but with known extreme bounds of the parameter values. Such inexactness or grey uncertainty in the model parameters can be addressed by the inexact or grey optimization models, representing the parameters as interval grey numbers. The research study presented in this thesis deals with the development of grey and fuzzy optimization models, and the combination of the two for water resources systems decision-making. Three grey fuzzy optimization models for waste load allocation, namely (i) Grey Fuzzy Waste Load Allocation Model (GFWLAM), (ii) two-phase GFWLAM and (iii) multiobjective GFWLAM, and a Grey Integer Programming (GIP) model for floodplain planning, are developed in this study.
The Grey Fuzzy Waste Load Allocation Model (GFWLAM) for water quality management of river system addresses uncertainty in the membership functions for imprecisely stated management goals of the Pollution Control Agency (PCA) and dischargers. To address the imprecision in fixing the boundaries of membership functions (also known as membership parameters), the membership functions themselves are treated as imprecise in the model and the membership parameters are expressed as interval grey numbers. The conflict between the fuzzy goals of PCA and dischargers is modeled using the concept of fuzzy decision, but because of treating the membership parameters as interval grey numbers, in the present study, the notion of ‘fuzzy decision’ is extended to the notion of ‘grey fuzzy decision’. A terminology ‘grey fuzzy decision’ is used to represent the fuzzy decision resulting from the imprecise membership functions. The model provides flexibility for PCA and dischargers to specify their aspirations independently, as the membership parameters for membership functions are interval grey numbers in place of a deterministic real number. In the solution, optimal fractional removal levels of the pollutants are obtained in the form of interval grey numbers. This enhances the flexibility and applicability in decision-making, as the decision-maker gets a range of optimal solutions for fixing the final decision scheme considering technical and economic feasibility of the pollutant treatment levels. The methodology is demonstrated with the case studies of a hypothetical river system and the Tunga-Bhadra river system in Karnataka, India.
Formulation of GFWLAM is based on the approach for solving fuzzy multiple objective optimization problem using max-min as the operator, which usually may not result in a unique solution. The two-phase GFWLAM captures all the alternative optimal solutions of the GFWLAM. The solution technique in the Phase 1 of two-phase GFWLAM is the same as that of GFWLAM. The Phase 2 maximizes upper bounds and minimizes lower bounds of decision variables, keeping the optimal value of goal fulfillment level same as obtained in the Phase 1. The two-phase GFWLAM gives the unique, widest, intervals of the optimal fractional removal levels of pollutant corresponding to the optimal value of goal fulfillment level. The solution increases the widths of interval-valued fractional removal levels of pollutants by capturing all the alternative optimal solutions and thus enhances the flexibility and applicability in decision-making. The model is applied to the case study of Tunga-Bhadra river system, which shows the existence of multiple solutions when the GFWLAM is applied to the same case study.
The width of the interval of optimal fractional removal level plays an important role in the GFWLAM, as more width in the fractional removals implies a wider choice to the decision-makers and more applicability in decision-making. The multiobjective GFWLAM maximizes the width of the interval-valued fractional removal levels for providing a latitude in decision-making and minimizes the width of goal fulfillment level for reducing the system uncertainty. The multiobjective GFWLAM gives a new methodology to get a satisfactory deterministic equivalent of a grey fuzzy optimization problem, using the concept of acceptability index for a meaningful ranking between two partially or fully overlapping intervals. The resulting multiobjective optimization model is solved by fuzzy multiobjective optimization technique. The consistency of the solution is verified by solving the problem with fuzzy goal programming technique. The multiobjective GFWLAM avoids intermediate submodels unlike GFWLAM, so that the solution from a single deterministic equivalent of the GFWLAM adequately covers all possible situations. Although the solutions obtained from multiobjective GFWLAM provide more flexibility than those of the GFWLAM, its application is limited to grey fuzzy goals expressed by linear imprecise membership functions only, whereas GFWLAM has the capability to solve the model with any monotonic nonlinear imprecise membership functions also. The methodology is demonstrated with the case studies of a hypothetical river system and the Tunga-Bhadra river system in Karnataka, India.
The Grey Integer Programming (GIP) model for floodplain planning is based on the floodplain planning model developed by Lund (2002), to identify an optimal mix of flood damage reduction options with probabilistic flood descriptions. The model demonstrates how the uncertainty of various input parameters in a floodplain planning problem can be modeled using interval grey numbers in the optimization model. The GIP model for floodplain planning does not replace a post-optimality analysis (e.g., sensitivity analysis, dual theory, parametric programming, etc.), but it provides additional information for interpretation of the optimal solutions. The results obtained from GIP model confirm that the GIP is a useful technique for interpretation of the solutions particularly when a number of potential feasible measures are available in a large scale floodplain planning problem. Though the present study does not directly compare the GIP technique with sensitivity analysis, the results indicate that the rigor and extent of post-optimality analyses may be reduced with the use of GIP for a large scale floodplain planning problem. Application of the GIP model is demonstrated with the hypothetical example as presented in Lund (2002).